Mellin amplitudes for dual conformal integrals

  • Miguel F. Paulos
  • Marcus Spradlin
  • Anastasia Volovich
Open Access
Article

Abstract

Motivated by recent work on the utility of Mellin space for representing conformal correlators in AdS/CFT, we study its suitability for representing dual conformal integrals of the type which appear in perturbative scattering amplitudes in super-Yang-Mills theory. We discuss Feynman-like rules for writing Mellin amplitudes for a large class of integrals in any dimension, and find explicit representations for several familiar toy integrals. However we show that the power of Mellin space is that it provides simple representations even for fully massive integrals, which except for the single case of the 4-mass box have not yet been computed by any available technology. Mellin space is also useful for exhibiting differential relations between various multi-loop integrals, and we show that certain higher-loop integrals may be written as integral operators acting on the fully massive scalar n-gon in n dimensions, whose Mellin amplitude is exactly 1. Our chief example is a very simple formula expressing the 6-mass double box as a single integral of the 6-mass scalar hexagon in 6 dimensions.

Keywords

Scattering Amplitudes Conformal and W Symmetry Duality in Gauge Field Theories AdS-CFT Correspondence 

References

  1. [1]
    L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills Theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    R. Roiban, M. Spradlin and A. Volovich eds., Special issue: Scattering amplitudes in gauge theories: progress and outlook, J. Phys. A 44 (2011) 450301.Google Scholar
  5. [5]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  6. [6]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural Language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  8. [8]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    D. Nandan, A. Volovich and C. Wen, On Feynman rules for Mellin amplitudes in AdS/CFT, JHEP 05 (2012) 129 [arXiv:1112.0305] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, arXiv:1112.4845 [INSPIRE].
  12. [12]
    J. Polchinski, S matrices from AdS space-time, hep-th/9901076 [INSPIRE].
  13. [13]
    L. Susskind, Holography in the flat space limit, hep-th/9901079 [INSPIRE].
  14. [14]
    S.B. Giddings, The Boundary S matrix and the AdS to CFT dictionary, Phys. Rev. Lett. 83 (1999) 2707 [hep-th/9903048] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  15. [15]
    T. Okuda and J. Penedones, String scattering in flat space and a scaling limit of Yang-Mills correlators, Phys. Rev. D 83 (2011) 086001 [arXiv:1002.2641] [INSPIRE].ADSGoogle Scholar
  16. [16]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-matrix, arXiv:1111.6972 [INSPIRE].
  17. [17]
    S. Raju, New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, arXiv:1201.6449 [INSPIRE].
  18. [18]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \(\mathcal{N} = 4\) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    V.A. Smirnov, Springer tracts in modern physics. Vol. 211: Evaluating Feynman Integrals, Springer, Heidelberg Germany (2004).Google Scholar
  25. [25]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    Z. Bern, J. Carrasco, H. Johansson and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].MathSciNetADSGoogle Scholar
  28. [28]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar \(\mathcal{N} = 4\) SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A.B. Goncharov, Polylogarithms and Motivic Galois groups, Proc. Symp. Pure Math. 55 (1994) 43.MathSciNetCrossRefGoogle Scholar
  32. [32]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076].MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math J. 128 (2005) 209 [math/0208144].MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    V. Del Duca, C. Duhr and V.A. Smirnov, The massless hexagon integral in D = 6 dimensions, Phys. Lett. B 703 (2011) 363 [arXiv:1104.2781] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in \(\mathcal{N} = 4\) SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    V. Del Duca, C. Duhr and V.A. Smirnov, The one-loop one-mass hexagon integral in D = 6 dimensions, JHEP 07 (2011) 064 [arXiv:1105.1333] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    V. Del Duca et al., The one-loop six-dimensional hexagon integral with three massive corners, Phys. Rev. D 84 (2011) 045017 [arXiv:1105.2011] [INSPIRE].ADSGoogle Scholar
  40. [40]
    M. Spradlin and A. Volovich, Symbols of one-loop integrals from mixed Tate motives, JHEP 11 (2011) 084 [arXiv:1105.2024] [INSPIRE].Google Scholar
  41. [41]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    P. Heslop and V.V. Khoze, Wilson loops @ 3-loops in special kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, arXiv:1110.0458 [INSPIRE].
  44. [44]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in \(\mathcal{N} = 4\) super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar \(\mathcal{N} = 4\) super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    A. Prygarin, M. Spradlin, C. Vergu and A. Volovich, All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology, Phys. Rev. D 85 (2012) 085019 [arXiv:1112.6365] [INSPIRE].ADSGoogle Scholar
  47. [47]
    A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in \(\mathcal{N} = 4\) SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, arXiv:1203.0454 [INSPIRE].
  49. [49]
    S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    N. Usyukina and A.I. Davydychev, An approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    J.M. Drummond and J.M. Henn, Simple loop integrals and amplitudes in \(\mathcal{N} = 4\) SYM, JHEP 05 (2011) 105 [arXiv:1008.2965] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [INSPIRE].
  54. [54]
    P.A.M. Dirac, n Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  55. [55]
    S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].ADSGoogle Scholar
  56. [56]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    K. Symanzik, On calculations in conformal invariant field theories, Lett. Nuovo Cim. 3 (1972) 734 [INSPIRE].CrossRefGoogle Scholar
  59. [59]
    S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, arXiv:1101.4163 [INSPIRE].
  60. [60]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    L. Mason and D. Skinner, Amplitudes at Weak Coupling as Polytopes in AdS 5, J. Phys. A 44 (2011) 135401 [arXiv:1004.3498] [INSPIRE].MathSciNetADSGoogle Scholar
  62. [62]
    M. Paulos, in preparation.Google Scholar

Copyright information

© SISSA 2012

Authors and Affiliations

  • Miguel F. Paulos
    • 1
  • Marcus Spradlin
    • 2
  • Anastasia Volovich
    • 2
  1. 1.Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589Université Pierre et Marie CurieParis Cedex 05France
  2. 2.Department of PhysicsBrown UniversityProvidenceU.S.A.

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